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Measurement of the second order molecular hyperpolarizability of C60− by nondegenerate four-wave mixing
25 April 1997
CHEMICAL PHYSICS LETTERS ELSEVIER
Chemical PhysicsLetters 269 (1997) 79-84
Measurement of the second order molecular hyperpolarizability of C 60 by nondegenerate four-wave mixing Robert Lascola, John C. Wright Department of Chemistry, 1101 UniversityAvenue, Universityof Wisconsin, Madison, W153706, USA
Received 2 January 1997
Abstract We have determined the second order molecular hyperpolarizability 3, for the charged species C6o by use of coherent anti-Stokes Raman spectroscopy (CARS) vibrational lineshape analysis. We find y is 2.4 (+ 1.0) × 10-3s esu, a value that is > 65 times larger than the corresponding value for the neutral species and is comparable to values for highly conjugated organic polymers like polydiacetylene. The observed increase in ~/ with addition of charge to the fullerene cage has encouraging implications for the use of charge-transfer and endohedral fullerene complexes in future photonics devices.
1. Introduction Fullerenes were initially considered promising candidates for nonlinear optical materials because initial measurements showed a very large second order molecular hyperpolarizability, 3', for C6o [1]. Later measurements did not confirm the large nonlinearity but neither did they agree with each other [2,3]. Nonlinear measurements are plagued by experimental difficulties and even differences in how the nonlinearities are defined. Even recent measurements of 3" range from 10 - 3 0 tO < 4 × 10 -35 e s u . In this Letter, definitive values for 3' are obtained by the coherent anti-Stokes Raman spectroscopy (CARS) lineshape analysis method of Levenson and Bloembergen [4] where one fits the dispersive lineshape caused by interference between the unknown nonlinearity and a Raman resonance of a standard like benzene. This procedure is well accepted and has been shown to provide accurate values for 3". It was used to measure C6o and it was found that the 3' value is too small to measure in saturated solutions
(3'(C6o)< 3.7 × 10 -35 esu [3]). In this Letter we report the measurement of y for the fullerene anion C/,0 obtained from CARS lineshape analysis. Our value, Y~lll = 2.4(-t- 1.0) × 10 -33 e s u , is comparable to the nonlinearities of organic polymers and indicates that charged fullerenes and metallofullerenes with excess charge on the fullerene cage have better potential for nonlinear materials than neutral fullerenes.
2. Experimental Solutions of C~o in tetrahydrofuran (THF) are made under inert atmosphere conditions from pure C60 (99.99%, SES) following the chemical reduction method of Boulas et al, [5]. The C~o preparation and purity is confirmed by near-IR spectroscopy [5,6]. The experimental apparatus for measuring CARS spectral iineshapes has been described previously [7,8]. The two excitation beams are from excimer-
0009-2614/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved. Pll S0009-2614(97)00243-1
R. Lascola, J.C. Wright/Chemical Physics Letters 269 (1997) 79-84
80
pumped (308 nm, 10 ns pulse width, 10 Hz repetition rate) dye lasers; to! is fixed at 22100 cm -1, while to2 is scanned so that t o ! - to2 ranges from 1100-800 cm -j. The angle between the two beams ( = 0.7 °) is adjusted for phase matching to maximize the signal intensity at the THF Raman resonance (914 cm-l). Total dye laser powers are less than 200 ixJ per pulse. Near-IR spectra confirm that the solutions do not degrade during the experiments. The accuracy of the method has been verified for both transparent and absorbing samples. The THF 914 cm -j Raman resonant, TR, and nonresonant, TN, values are first calibrated as secondary standards by measuring the values against benzene, a primary standard for 3'. This resonance is chosen because its nonlinearity is relatively weak and comparable to the C6o contribution at the concentrations used in this experiment; the method's precision is optimized under this condition. Pure THF and T H F / C 6 H 6 mixtures are studied in standard 1 cm cuvettes where window contributions are negligible, as they are outside the confocal distance of the focused laser beams (0.6 cm). The nonresonant and vibrationally resonant susceptibilities of the THF solvent are measured by the lineshape analysis of CARS spectra of THF/benzene mixtures. The measurements of the absorbing C~o solutions are taken with an optical cell with a continuously adjustable and thin pathlength ( 5 0 - > 1000 p.m) and l l 0 Ixm thin borosilicate windows [9]. The total thickness of the sample and windows (typically 350 Ixm) is smaller than the confocal distance, and thus the windows contribute to the CARS signal. The C~o concentrations were 4.5, 2.2, 1.3, and 0.0 mM.
3. Theory The sample's nonlinear polarization in a CARS experiment with parallel polarized lasers is given by P(to3
= 2 t o l -- t o 2 )
= X ~ 1( to3 ; tol , - -
to2,to,)E, EzE,,
(1)
where ,,(3) a l ~ l l = N F T ~ I I , ,,(3) alibi is the bulk third order nonlinear susceptibility, N is the number density of the nonlinear chromophores, F is the field correction factor [(n 2 + 2)/3] 4, n is the index of refraction, and
the E i are the electric fields associated with the lasers at frequency °Jr X~]l includes contributions from the sample ( Xs(3)) and (if appropriate) the windows (X(w3)). In the C6a/THF experiments, Xs(3) has contributions from the THF Raman line at 914 c m - l (X(R3)) and electronic contributions from the THF and C~0 (X(B3)): ,,(3) Xs --( AIR 3 "1"-)X (3) m. OR--
AReiOR
_
( t o , - °)2) --iFR
+ AB eiOB ,
(2)
where A i and 0 i represent the magnitude and complex phase angle from electronic contributions and, g2R, and F a are frequency and linewidth of a Raman transition [10]. The electronic contributions are assumed to have negligible frequency dependences although they may include nearby or even resonant states [11]. The output electric field E 3 depends not only on the X <3) values but also on the absorbance of the input and output beams by the sample. In our experiments, the signal field is given by [12]
E, = e-(' J2'°:S(Ms
3' + Mw
where e (2i4~s-A t~Ls) -- 1
Ms = e-i't'w 2 ( A a + iAks) '
(3)
and sin 4'w - Mw -- Ak w
( e [2i('bs+4'w)- A°tLs] -k- 1 ) ,
where Act = ½(2a I + a z - a 3) ( a i are the intensity absorption coefficients of the solution at frequency to/), Ak w is the wave vector mismatch and thw = AkwLw/2 is the phase-matching argument within the window of thickness L w (and similarly for A k s and ~bs within the sample of thickness Ls). The output intensity is proportional to IE3Iz so the contributions from the different solution components that add at the amplitude level will interfere at the intensity level because of the cross-terms. This interference gives rise to the dispersive Raman lineshape observed in the data.
R. Lascola, J. C. Wright/Chemical Physics Letters 269 (1997) 79-84 4. R e s u l t s a n d d i s c u s s i o n
Details of the T H F / C r H 6 experiments will be presented elsewhere [9]; results for THF are shown in Table 1. We will concentrate on C~,JTHF mixtures. Twenty-six spectra of C~o solutions with different concentrations (4.5, 2.2, 1.3, 0.0 mM) were taken on several occasions. Fig. 1 shows a typical lineshape and theoretical fit for a 4.5 mM C~0 sample, and demonstrates the lineshape change compared to a pure THF solution. For each spectrum, L s, Lw, Aw, Aks, ~bs, Akw, thw and all tx are independently measured to within 10% and fixed during the fit. X O) is set equal to the literature value for borosilicate [13] and is assumed to be real. The parameters AR/AB, ~'~R' Fa, AR/Mw, and (0 R 0 B) are determined from the fits. Their values, averaged over all scans for a given C~o concentration, are given in Table 2. Error estimates for all of the parameters are determined from the Marquardt least squares regression analysis [14] of each individual spectrum. Independent estimates are obtained from
81
the standard deviation of values from all fitted spectra. The two estimates are consistent and the larger of the two are tabulated. Although there are 5 parameters, each controls distinct spectral features, so their values are well defined in the fitting procedures. This is reflected by the low cross-correlation coefficients generated by the fits (generally within +0.4). Note that the window contribution is not large compared with the total signal. In addition, less satisfactory fits were obtained when the values for Ls, Lw, Mw, Aks, ths, Akw, ~bw, X O) and all a l, were changed from their measured or accepted values. The fitting to the individual spectra provides values for the total background nonlinearity and it includes contributions from both the THF and the C~o. In order to define the contributions from C6o in a mixture, both cases of a mixture, the relative contributions of the two components in the solution are varied by changing the ratio of background nonlinearity to the THF Raman nonlinearity at 914 cm -1 (R m - X(B3)/X(~3)(914c m - l ) ) is measured for the series of mixtures and compared to the ratio
• 0.75
1.0
0.25 ......
,,.,~
O O O L ~ = ~ ; . ~ 4
: ~ ~:~::::~:'~
.... ~ "~':' '~Ei~:~)~:~ !!!i~' :~~,' : ' , ..~ .z ~.::..%:::,' .:.:..~ ' ~ b ..i ,.i:, ,, ~:~ i,'...,, !.!' ..:.,'..,~'~:....... '/:
o
! [Cool ( m a )
~
¢.-
, i~,~ i :::. : .i: , : : : i i ~ (b)
_>, 0.5 ¢/) r-
c-
(a)
0.0 0.2 I ~- 0.0 -0.2
r 800
850
900
950
1000
c%-co2 (cm") Fig. 1. Top: Dotted lines: normalized spectra of (a) pure THF and (b) 4.5 mM C~o in THF. Solid lines: results of spectral fits, as discussed in the text. Bottom: Residuals p for spectrum (b), as a fraction of total signal. Inset: C~o concentration dependence of the background nonlinearity. The line represents a linear least-squares fit to the points.
82
R. Lascola, J. C Wright/Chemical Physics Letters 269 (1997) 79-84
Table 1 Nonlinear susceptibilities ( XO)) and hyperpolarizabilities (3') for THF and C~-o THF C~ X~3) (cm3/,erg) 1.4 (0.2)× 10- 14 -- 10- i i Ya (esu) 6.0 (0.7)× l0 -37 2.4 (l.0)× l0 -33 XtR3) (cm3/'erg) 2.6 (0.2) x 10- ~4 _ YR (esn) l.l (0.1)× l0 -36 -The subscript 'B' indicates the background (electronic) nonlinearity, and 'R' indicates the vibrational resonance at 914 cm-i (for THF). All quantities represent the (l l 11) tensor component. X and y are related by X O ) = f ~ f ~ 2 f Z N ( y ) where N is the number density, f,o = (n2(t°) + 2)/`3 is the Lorenz field correction factor, and n(to) is the index of refraction. See text for details of the calculation of X(a3) for C6o. All uncertainties in parentheses represent 2 o-. measured in the pure T H F standard, R s. Since the total nonlinearity depends linearly on the individual contributions, Yc~o, can be extracted from the relationship R m -R s =
Tc~o Ncgo
,
(4)
'YTHFNTHF where Tm is the hyperpolarizability and N m is the number density of species m. Note that the local field correction factor does not enter since that factor is the same for the mixture and the standard. We have previously demonstrated [3] the ability of this procedure to make accurate determinations of hyperpolarizabilities from solutions with multiple components.
The data for the concentration-dependent ratios is shown in Table 2 and the results of the fit of that data to Eq. 4 are shown on the insert in Fig. 1. Several points should be noted. The excellent linearity of the background with concentration is evidence for the ability of the model to separate the effects of sample absorption (which limits the pathlength of four wave mixing in the sample while not affecting the process in the windows) from the increase in nonresonant background due to the increased C60 present [15]. In addition, the y for THF measured from the y-intercept matches the value measured in a thick cell where window contributions were negligible. This agreement provides evidence that the model properly accounts for the small window contribution. The 3' for C60 is determined from the slope of the concentration dependence and is insensitive to the absorbance, window contribution, and pathlengths. A detailed study of the error analysis of these measurements will be published elsewhere [9]. W e report the electronic molecular hyperpolarizability y(C60) = 2.4 ( + 1.0) × 10 -33 esu. (The imaginary component of y ( C ~ o ) is zero within experimental uncertainty.) Our value for Y(C60) is at least 65 times greater than the corresponding value for the neutral species measured at the same wavelength. Somewhat smaller increases have been observed experimentally for C6o containing charge-transfer complexes [16] and predicted for the isoelectronic species C59 N [17]. There are two factors that can explain the increase. First, neutral C6o has a filled 4h u orbital
Table 2 Averaged fit results for solutions of different C~o concentrations [C~o] (mM) AR/A a Ar/,M w OR - 0a (°) O R (cm- ~) F R (cm-J) Rm a Rs b
0.0
1.3
11.4 (1.3) 18.0 (0.6) 4.0 (4.4) 912.3 (0.5) 6.3 (0.2) 0.55 (0.08) 0.54 (0.4)
7.9 (0.5) 18.1 (2.6) 0.8 (3.0) 911.9 (0.4) 6.3 (0.9) 0.79 (0.06) -
a M e a s u r e d in 1 c m thick cuvette. b M e a s u r e d in 1 3 0 p.m thick s a m p l e
2.2
4.5
6.8 (1.2) 19.1 (1.2)
4.8 (0.5) 18.1 (1.4)
22 ( 7 ) 912.5 (0.3) 6.7 (0.4) 1.01 (0.14) -
911.4 (0.7) 6.3 (0.5) 1.35 (0.14) -
- 3.1 ( 9 . 2 )
cell. Rs is determined from fits of spectra of pure THF, as discussed in the Experimental section. The standard deviations of the measurements are given in parentheses.
R. Lascola, J. C Wright/Chemical Physics Letters 269 (1997) 79-84
and allowed transitions occur at higher energies. The additional electron in the C7,o anion enters an empty 5tl, orbital and there are many additional allowed transitions that begin in the near infrared. These allowed transitions would enhance the nonlinearity through their higher oscillator strengths and proximity. Secondly, Hess and co-workers [11] have shown that the strongest coherent pathways destructively interfere in C6o and the observed y is the result of differences between large competing effects. This situation makes y susceptible to changes in the radiative coupling strengths that can make large changes in the balance between coherent pathways. If the cancellation between pathways is eliminated, large nonlinearities may result. This measurement of the hyperpolarizability of the anion suggests that charged fullerenes can have bulk nonlinearities that are comparable to the best organic materials. High concentrations of fullerenes have been incorporated into crystalline salts, clathrate compounds, and polymers. A typical concentration for fulleride salts and clathrate compounds is 1021 cm-3 [ 18]. This concentration would give a susceptibility of A"~3)= 10 -11 cm3/erg assuming n = 2 . 0 . One of the best nonlinear materials is the polydiacetylene-based polymer PTS which has a susceptibility of X O) = 5 × 10-10 cm3/erg [19]. Although this value is 50 × higher than our value for C~o, it is comparable and it is likely the C~o X O) can be optimized by derivatization, choice of matrix, and reduction to lower valence states. It will also be interesting to examine lanthanide endohedral fullerene complexes. The lanthanide ions within the fullerene cage are divalent or trivalent cations and require a charged anionic cage as the charge compensation [20]. The endohedral fullerene cage's electronic structure would be similar to the fulleride ions and might be excellent candidates for making nonlinear materials.
Acknowledgements We would like to acknowledge Professor Robert West for the use of Schlenk-lines and a glove box.
83
This research is supported by the National Science Foundation under grant DMR-9632293. RL acknowledges the support of an American Chemical Society, Division of Analytical Chemistry Fellowship sponsored by Eastman Chemical Company.
References [1] W.J. Blau, H.J. Byrne, D.J. Cardin, T.J. Dennis, J.P. Hare, H.W. Kroto, R. Taylor, D.R.M. Walton, Phys. Rev. Len. 67 (1991) 1423. [2] F. Kajzar, C. Taliani, R. Zamboni, S. Rossini, R. Danieli, Synth. Met. 77 (1996) 257; D. Zhou, L. Gan, C. Luo, H. Tang, C. Huang, Z. Liu, Z. Wu, X. Zhao, X. Xia, S. Zhang, F. Sun, Z. Xia, Y. Zou, Chem. Phys. Lett. 235 (1995) 548; N. Tang, J,P. Partanen, R.W. Hellwarth, R.J. Knize, Phys. Rev. B 48 (1993) 8404. [3] L. Geng, J.C. Wright, Chem. Phys. Lett. 249 (1996) 105. [4] M.D. Levenson, N, Bloembergen, J. Chem. Phys. 60 (1974) 1323. [5] P. Boulas, R. Subramanian, W. Kutner, M.T. Jones, K.M. Kadish, J. Electrochem. Soc. 140 (1993) L130. [6] D.R. Lawson, D.L. Feldman, C.A. Foss, PK. Dorhout, C.M. Elliot, C.R. Martin, B. Parkinson, J. Electrochem. Soc. 139 (1992) L68. [7] J.C. Wright, R.J. Carlson, G.B. Hurst, J.K. Steehler, M.T. Reibe, B.B. Price, D.C. Nguyen, S.H. Lee, Int. Rev. Phys. Chem. 10 (1991) 349. [8] S.H. Lee, J.K. Steehler, D.C. Nguyen, J.C. Wright, Appl. Spectrosc. 39 (1985) 243. [9] R. Lascola, J. C. Wright, to be published. [10] W.M. Tolles, J.W, Nibler, J.R. McDonald, A.B. Harvey, Appl. Spectrosc. 31 (1977) 253. [1 I] B.C. Hess, D.V. Bowersox, S.H. Mardirosian, L.D. Unterberger, Chem. Phys. Lett. 248 (1996) 141. [12] T.A.H.M. Scholten, G.W. Lucassen, E. Koelewijn, F.FM. de Mul, J. Greve, J. Raman Spectrosc. 20 (1989) 503. [13] M.D. Levenson, IEEE J. Quantum Electron. QE-10(2) (1974) 110. [14] P.R. Bevington, D.K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, 2nd (Ed.), McGraw-Hill, New York, 1989, p. 161-163. [15] R.J. Carlson, J.C. Wright, Appi. Spectrosc. 43 (1989) 1195. [16] Z.H. Kafafi, S.R. Flom, H.W. Sarkas, R.G.S. Pong, C.D. Merritt, F.J. Bartoli, Proc. SPIE - Int. Soc. Opt. Eng. 2025 (1993) 134. [17] J. Dong, J. Jiang, J. Yu, Z.D. Wang, D.Y. Xing, Phys. Rev. B 52 (1995) 9066. [18] O. Ermer, Heir. Chim. Acta 74 (1991) 1339; P. Paul, Z. Xie, R. Bau, P.D.W. Boyd, C.A. Reed, J. Am. Chem. Soc. 116 (1994) 4145; P. W. Stephens, D. Cox, J.W. Lauher, L. Mihaly, J.B. Wiley, P.-M. Allemand, A. Hirsch, K. Holczer,
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R. Lascola, J. C. Wright/Chemical Physics Letters 269 (1997) 79-84
Li, J.D. Thompson, F. Wudl, Nature 355 (1992) 331; A. lzuoka, T. Tachikawa, T. Sugawara, Y. Suzuki, M. Konno, Y, Saito, H. Shinohara, J. Chem. Soc. Chem. Commun. (1992) 1472. [19] G.M. Carter, Y.J. Chen, M.F. Rubner, D.J. Sandman, M.K. Thakur, S.K. Tripathy, in: Nonlinear Optical Properties of
Organic Molecules and Crystals, Vol. 2, D.S. Chemla, J. Zyss (Eds.), Academic Press, New York, 1987, p. 85. [20] D.M. Poirier, M. Knupfer, J.H. Weaver, W. Andreoni, K. Laasonen, M. Parinello, D.S. Bethune, K. Kikuchi, Y. Achiba, Phys. Rev. B 49 (1994) 17403.
CHEMICAL PHYSICS LETTERS ELSEVIER
Chemical PhysicsLetters 269 (1997) 79-84
Measurement of the second order molecular hyperpolarizability of C 60 by nondegenerate four-wave mixing Robert Lascola, John C. Wright Department of Chemistry, 1101 UniversityAvenue, Universityof Wisconsin, Madison, W153706, USA
Received 2 January 1997
Abstract We have determined the second order molecular hyperpolarizability 3, for the charged species C6o by use of coherent anti-Stokes Raman spectroscopy (CARS) vibrational lineshape analysis. We find y is 2.4 (+ 1.0) × 10-3s esu, a value that is > 65 times larger than the corresponding value for the neutral species and is comparable to values for highly conjugated organic polymers like polydiacetylene. The observed increase in ~/ with addition of charge to the fullerene cage has encouraging implications for the use of charge-transfer and endohedral fullerene complexes in future photonics devices.
1. Introduction Fullerenes were initially considered promising candidates for nonlinear optical materials because initial measurements showed a very large second order molecular hyperpolarizability, 3', for C6o [1]. Later measurements did not confirm the large nonlinearity but neither did they agree with each other [2,3]. Nonlinear measurements are plagued by experimental difficulties and even differences in how the nonlinearities are defined. Even recent measurements of 3" range from 10 - 3 0 tO < 4 × 10 -35 e s u . In this Letter, definitive values for 3' are obtained by the coherent anti-Stokes Raman spectroscopy (CARS) lineshape analysis method of Levenson and Bloembergen [4] where one fits the dispersive lineshape caused by interference between the unknown nonlinearity and a Raman resonance of a standard like benzene. This procedure is well accepted and has been shown to provide accurate values for 3". It was used to measure C6o and it was found that the 3' value is too small to measure in saturated solutions
(3'(C6o)< 3.7 × 10 -35 esu [3]). In this Letter we report the measurement of y for the fullerene anion C/,0 obtained from CARS lineshape analysis. Our value, Y~lll = 2.4(-t- 1.0) × 10 -33 e s u , is comparable to the nonlinearities of organic polymers and indicates that charged fullerenes and metallofullerenes with excess charge on the fullerene cage have better potential for nonlinear materials than neutral fullerenes.
2. Experimental Solutions of C~o in tetrahydrofuran (THF) are made under inert atmosphere conditions from pure C60 (99.99%, SES) following the chemical reduction method of Boulas et al, [5]. The C~o preparation and purity is confirmed by near-IR spectroscopy [5,6]. The experimental apparatus for measuring CARS spectral iineshapes has been described previously [7,8]. The two excitation beams are from excimer-
0009-2614/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved. Pll S0009-2614(97)00243-1
R. Lascola, J.C. Wright/Chemical Physics Letters 269 (1997) 79-84
80
pumped (308 nm, 10 ns pulse width, 10 Hz repetition rate) dye lasers; to! is fixed at 22100 cm -1, while to2 is scanned so that t o ! - to2 ranges from 1100-800 cm -j. The angle between the two beams ( = 0.7 °) is adjusted for phase matching to maximize the signal intensity at the THF Raman resonance (914 cm-l). Total dye laser powers are less than 200 ixJ per pulse. Near-IR spectra confirm that the solutions do not degrade during the experiments. The accuracy of the method has been verified for both transparent and absorbing samples. The THF 914 cm -j Raman resonant, TR, and nonresonant, TN, values are first calibrated as secondary standards by measuring the values against benzene, a primary standard for 3'. This resonance is chosen because its nonlinearity is relatively weak and comparable to the C6o contribution at the concentrations used in this experiment; the method's precision is optimized under this condition. Pure THF and T H F / C 6 H 6 mixtures are studied in standard 1 cm cuvettes where window contributions are negligible, as they are outside the confocal distance of the focused laser beams (0.6 cm). The nonresonant and vibrationally resonant susceptibilities of the THF solvent are measured by the lineshape analysis of CARS spectra of THF/benzene mixtures. The measurements of the absorbing C~o solutions are taken with an optical cell with a continuously adjustable and thin pathlength ( 5 0 - > 1000 p.m) and l l 0 Ixm thin borosilicate windows [9]. The total thickness of the sample and windows (typically 350 Ixm) is smaller than the confocal distance, and thus the windows contribute to the CARS signal. The C~o concentrations were 4.5, 2.2, 1.3, and 0.0 mM.
3. Theory The sample's nonlinear polarization in a CARS experiment with parallel polarized lasers is given by P(to3
= 2 t o l -- t o 2 )
= X ~ 1( to3 ; tol , - -
to2,to,)E, EzE,,
(1)
where ,,(3) a l ~ l l = N F T ~ I I , ,,(3) alibi is the bulk third order nonlinear susceptibility, N is the number density of the nonlinear chromophores, F is the field correction factor [(n 2 + 2)/3] 4, n is the index of refraction, and
the E i are the electric fields associated with the lasers at frequency °Jr X~]l includes contributions from the sample ( Xs(3)) and (if appropriate) the windows (X(w3)). In the C6a/THF experiments, Xs(3) has contributions from the THF Raman line at 914 c m - l (X(R3)) and electronic contributions from the THF and C~0 (X(B3)): ,,(3) Xs --( AIR 3 "1"-)X (3) m. OR--
AReiOR
_
( t o , - °)2) --iFR
+ AB eiOB ,
(2)
where A i and 0 i represent the magnitude and complex phase angle from electronic contributions and, g2R, and F a are frequency and linewidth of a Raman transition [10]. The electronic contributions are assumed to have negligible frequency dependences although they may include nearby or even resonant states [11]. The output electric field E 3 depends not only on the X <3) values but also on the absorbance of the input and output beams by the sample. In our experiments, the signal field is given by [12]
E, = e-(' J2'°:S(Ms
3' + Mw
where e (2i4~s-A t~Ls) -- 1
Ms = e-i't'w 2 ( A a + iAks) '
(3)
and sin 4'w - Mw -- Ak w
( e [2i('bs+4'w)- A°tLs] -k- 1 ) ,
where Act = ½(2a I + a z - a 3) ( a i are the intensity absorption coefficients of the solution at frequency to/), Ak w is the wave vector mismatch and thw = AkwLw/2 is the phase-matching argument within the window of thickness L w (and similarly for A k s and ~bs within the sample of thickness Ls). The output intensity is proportional to IE3Iz so the contributions from the different solution components that add at the amplitude level will interfere at the intensity level because of the cross-terms. This interference gives rise to the dispersive Raman lineshape observed in the data.
R. Lascola, J. C. Wright/Chemical Physics Letters 269 (1997) 79-84 4. R e s u l t s a n d d i s c u s s i o n
Details of the T H F / C r H 6 experiments will be presented elsewhere [9]; results for THF are shown in Table 1. We will concentrate on C~,JTHF mixtures. Twenty-six spectra of C~o solutions with different concentrations (4.5, 2.2, 1.3, 0.0 mM) were taken on several occasions. Fig. 1 shows a typical lineshape and theoretical fit for a 4.5 mM C~0 sample, and demonstrates the lineshape change compared to a pure THF solution. For each spectrum, L s, Lw, Aw, Aks, ~bs, Akw, thw and all tx are independently measured to within 10% and fixed during the fit. X O) is set equal to the literature value for borosilicate [13] and is assumed to be real. The parameters AR/AB, ~'~R' Fa, AR/Mw, and (0 R 0 B) are determined from the fits. Their values, averaged over all scans for a given C~o concentration, are given in Table 2. Error estimates for all of the parameters are determined from the Marquardt least squares regression analysis [14] of each individual spectrum. Independent estimates are obtained from
81
the standard deviation of values from all fitted spectra. The two estimates are consistent and the larger of the two are tabulated. Although there are 5 parameters, each controls distinct spectral features, so their values are well defined in the fitting procedures. This is reflected by the low cross-correlation coefficients generated by the fits (generally within +0.4). Note that the window contribution is not large compared with the total signal. In addition, less satisfactory fits were obtained when the values for Ls, Lw, Mw, Aks, ths, Akw, ~bw, X O) and all a l, were changed from their measured or accepted values. The fitting to the individual spectra provides values for the total background nonlinearity and it includes contributions from both the THF and the C~o. In order to define the contributions from C6o in a mixture, both cases of a mixture, the relative contributions of the two components in the solution are varied by changing the ratio of background nonlinearity to the THF Raman nonlinearity at 914 cm -1 (R m - X(B3)/X(~3)(914c m - l ) ) is measured for the series of mixtures and compared to the ratio
• 0.75
1.0
0.25 ......
,,.,~
O O O L ~ = ~ ; . ~ 4
: ~ ~:~::::~:'~
.... ~ "~':' '~Ei~:~)~:~ !!!i~' :~~,' : ' , ..~ .z ~.::..%:::,' .:.:..~ ' ~ b ..i ,.i:, ,, ~:~ i,'...,, !.!' ..:.,'..,~'~:....... '/:
o
! [Cool ( m a )
~
¢.-
, i~,~ i :::. : .i: , : : : i i ~ (b)
_>, 0.5 ¢/) r-
c-
(a)
0.0 0.2 I ~- 0.0 -0.2
r 800
850
900
950
1000
c%-co2 (cm") Fig. 1. Top: Dotted lines: normalized spectra of (a) pure THF and (b) 4.5 mM C~o in THF. Solid lines: results of spectral fits, as discussed in the text. Bottom: Residuals p for spectrum (b), as a fraction of total signal. Inset: C~o concentration dependence of the background nonlinearity. The line represents a linear least-squares fit to the points.
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R. Lascola, J. C Wright/Chemical Physics Letters 269 (1997) 79-84
Table 1 Nonlinear susceptibilities ( XO)) and hyperpolarizabilities (3') for THF and C~-o THF C~ X~3) (cm3/,erg) 1.4 (0.2)× 10- 14 -- 10- i i Ya (esu) 6.0 (0.7)× l0 -37 2.4 (l.0)× l0 -33 XtR3) (cm3/'erg) 2.6 (0.2) x 10- ~4 _ YR (esn) l.l (0.1)× l0 -36 -The subscript 'B' indicates the background (electronic) nonlinearity, and 'R' indicates the vibrational resonance at 914 cm-i (for THF). All quantities represent the (l l 11) tensor component. X and y are related by X O ) = f ~ f ~ 2 f Z N ( y ) where N is the number density, f,o = (n2(t°) + 2)/`3 is the Lorenz field correction factor, and n(to) is the index of refraction. See text for details of the calculation of X(a3) for C6o. All uncertainties in parentheses represent 2 o-. measured in the pure T H F standard, R s. Since the total nonlinearity depends linearly on the individual contributions, Yc~o, can be extracted from the relationship R m -R s =
Tc~o Ncgo
,
(4)
'YTHFNTHF where Tm is the hyperpolarizability and N m is the number density of species m. Note that the local field correction factor does not enter since that factor is the same for the mixture and the standard. We have previously demonstrated [3] the ability of this procedure to make accurate determinations of hyperpolarizabilities from solutions with multiple components.
The data for the concentration-dependent ratios is shown in Table 2 and the results of the fit of that data to Eq. 4 are shown on the insert in Fig. 1. Several points should be noted. The excellent linearity of the background with concentration is evidence for the ability of the model to separate the effects of sample absorption (which limits the pathlength of four wave mixing in the sample while not affecting the process in the windows) from the increase in nonresonant background due to the increased C60 present [15]. In addition, the y for THF measured from the y-intercept matches the value measured in a thick cell where window contributions were negligible. This agreement provides evidence that the model properly accounts for the small window contribution. The 3' for C60 is determined from the slope of the concentration dependence and is insensitive to the absorbance, window contribution, and pathlengths. A detailed study of the error analysis of these measurements will be published elsewhere [9]. W e report the electronic molecular hyperpolarizability y(C60) = 2.4 ( + 1.0) × 10 -33 esu. (The imaginary component of y ( C ~ o ) is zero within experimental uncertainty.) Our value for Y(C60) is at least 65 times greater than the corresponding value for the neutral species measured at the same wavelength. Somewhat smaller increases have been observed experimentally for C6o containing charge-transfer complexes [16] and predicted for the isoelectronic species C59 N [17]. There are two factors that can explain the increase. First, neutral C6o has a filled 4h u orbital
Table 2 Averaged fit results for solutions of different C~o concentrations [C~o] (mM) AR/A a Ar/,M w OR - 0a (°) O R (cm- ~) F R (cm-J) Rm a Rs b
0.0
1.3
11.4 (1.3) 18.0 (0.6) 4.0 (4.4) 912.3 (0.5) 6.3 (0.2) 0.55 (0.08) 0.54 (0.4)
7.9 (0.5) 18.1 (2.6) 0.8 (3.0) 911.9 (0.4) 6.3 (0.9) 0.79 (0.06) -
a M e a s u r e d in 1 c m thick cuvette. b M e a s u r e d in 1 3 0 p.m thick s a m p l e
2.2
4.5
6.8 (1.2) 19.1 (1.2)
4.8 (0.5) 18.1 (1.4)
22 ( 7 ) 912.5 (0.3) 6.7 (0.4) 1.01 (0.14) -
911.4 (0.7) 6.3 (0.5) 1.35 (0.14) -
- 3.1 ( 9 . 2 )
cell. Rs is determined from fits of spectra of pure THF, as discussed in the Experimental section. The standard deviations of the measurements are given in parentheses.
R. Lascola, J. C Wright/Chemical Physics Letters 269 (1997) 79-84
and allowed transitions occur at higher energies. The additional electron in the C7,o anion enters an empty 5tl, orbital and there are many additional allowed transitions that begin in the near infrared. These allowed transitions would enhance the nonlinearity through their higher oscillator strengths and proximity. Secondly, Hess and co-workers [11] have shown that the strongest coherent pathways destructively interfere in C6o and the observed y is the result of differences between large competing effects. This situation makes y susceptible to changes in the radiative coupling strengths that can make large changes in the balance between coherent pathways. If the cancellation between pathways is eliminated, large nonlinearities may result. This measurement of the hyperpolarizability of the anion suggests that charged fullerenes can have bulk nonlinearities that are comparable to the best organic materials. High concentrations of fullerenes have been incorporated into crystalline salts, clathrate compounds, and polymers. A typical concentration for fulleride salts and clathrate compounds is 1021 cm-3 [ 18]. This concentration would give a susceptibility of A"~3)= 10 -11 cm3/erg assuming n = 2 . 0 . One of the best nonlinear materials is the polydiacetylene-based polymer PTS which has a susceptibility of X O) = 5 × 10-10 cm3/erg [19]. Although this value is 50 × higher than our value for C~o, it is comparable and it is likely the C~o X O) can be optimized by derivatization, choice of matrix, and reduction to lower valence states. It will also be interesting to examine lanthanide endohedral fullerene complexes. The lanthanide ions within the fullerene cage are divalent or trivalent cations and require a charged anionic cage as the charge compensation [20]. The endohedral fullerene cage's electronic structure would be similar to the fulleride ions and might be excellent candidates for making nonlinear materials.
Acknowledgements We would like to acknowledge Professor Robert West for the use of Schlenk-lines and a glove box.
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This research is supported by the National Science Foundation under grant DMR-9632293. RL acknowledges the support of an American Chemical Society, Division of Analytical Chemistry Fellowship sponsored by Eastman Chemical Company.
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